A burning building is both an immediate threat to folks inside the building, and can be a threat to life to folks outside the building. That would be silly, however there should be accountability paperwork outside. In addition, many fires are reported by alarm systems or folks calling for emergency help - they asked for such a thing. There should be accountability paperwork, signed off by a court, before they start investigating - and the investigator should be a trained public official, not an insurance investigator.
In home murder? Assuming they got an emergency call or the cop happened to hear screaming, etc - yes, they should go in. Once folks are safe, however, everything else should require a warrant and paperwork should be filled out regardless for accountability reasons.
The whole point with the courts being involved is that even if it is after the incident, it brings a certain amount of accountability. We can change things so that different folks rotate the resopnsibility, include folks like prosecutors and public defenders and other private lawyers, and some things will basically be rubber-stamped (like investigating a fire or after they stopped the murder in progress).
In some cases, it is better to pass a law stating that a specific thing is allowed without a warrant, however, so that the courts don't have to get involved. A breathalyzer, for example, isn't invasive and good for public safety. Drawing blood should require a warrant and a trained professional in a clean environment, however. That could change if we have tests that are the equivalent of a finger prick for blood sugar as the risks are lower - we just aren't there yet.